First Robin Eigenvalue of the $p$-Laplacian on Riemannian Manifolds

TL;DR

This paper establishes comparison theorems and bounds for the first Robin eigenvalue of the p-Laplacian on Riemannian manifolds, linking geometric properties to eigenvalue estimates, including limits to Dirichlet cases.

Contribution

It provides new eigenvalue comparison theorems and sharp bounds for the Robin eigenvalues of the p-Laplacian on Riemannian manifolds, extending known results to Robin boundary conditions.

Findings

Eigenvalue comparison theorems of Cheng type for $igenvalues

Sharp lower bounds for $igenvalues$ when $>0$

Bounds become upper bounds when $<0$

Abstract

We consider the first Robin eigenvalue $\l_p(M,\a)$ for the $p$-Laplacian on a compact Riemannian manifold $M$ with nonempty smooth boundary, with $\a \in \R$ being the Robin parameter. Firstly, we prove eigenvalue comparison theorems of Cheng type for $\l_p(M,\a)$. Secondly, when $\a>0$ we establish sharp lower bound of $\l_p(M,\a)$ in terms of dimension, inradius, Ricci curvature lower bound and boundary mean curvature lower bound, via comparison with an associated one-dimensional eigenvalue problem. The lower bound becomes an upper bound when $\a<0$. Our results cover corresponding comparison theorems for the first Dirichlet eigenvalue of the $p$-Laplacian when letting $\a \to +\infty$.